In the period strip (1), we have one solution z1subscriptz1z_{1} and one solution z2subscriptz2z_{2}, both obtained with the value n=0n0n=0 (except z2subscriptz2z_{2} in the case φ=-πφπ\varphi=-\pi with n=-1n1n=-1). In (1), the pointsz1subscriptz1z_{1} and z2subscriptz2z_{2} are situated symmetrically with respect the origin. In the cases w=1w1w=1 and w=-1w1w=-1, the equation (2) has double roots z=0z0z=0 and z=-πzπz=-\pi, respectively; then we may say that z1subscriptz1z_{1} and z2subscriptz2z_{2} coincide. Anyhow, we have the

Theorem. In every period strip, cosine attains any complex value at two points.

Example. The solution of the equation cos⁡z=2z2\cos{z}=2 is obtained from ei⁢z=2±3superscripteizplus-or-minus23e^{{iz}}=2\!\pm\!\sqrt{3}. In the period strip (1) we get

Since 2±3plus-or-minus232\!\pm\!\sqrt{3} are inverse numbers of each other, we have as result the purely imaginary numbersz=±i⁢ln⁡(2+3)zplus-or-minusi23z=\pm{i}\ln(2\!+\!\sqrt{3}).

From trigonometry, we know that the real zeros of cosine are the odd multiples of π2π2\displaystyle\frac{\pi}{2}; from these points, ±π2plus-or-minusπ2\displaystyle\pm\frac{\pi}{2} belong to the period strip (1). Thus ±π2plus-or-minusπ2\displaystyle\pm\frac{\pi}{2} are the only points of (1) where the cosine vanishes. Therefore, according to the preceding theorem, the well-known points

One can think all points of the zzz-plane to bear the corresponding value of cosine, and then one can translate the plane in the direction of the real axis the distanceπ2π2\displaystyle\frac{\pi}{2}; then the values of the sine have been placed to their correct places. So one has transferred also the above properties of cosine to sine.